$button will always hold either the value of the input named submit, or the string "default_value". I'm guessing this is a submit button, and that you have given it a value (other than 0), so this will NEVER be true, even if someone reaches the page without submitting the form
if (!$button)
The same line of reasoning goes for $search. If the user inputs nothing, $search will be set to "default_value", and later on you will tell the user they searched for "default_value", and also needlessly query the database
Replace "default_value" with false for $button, and replace it with '' (empty string) for $search. If you want to supply a default string to search for, do so in the value attribute of the text input in the form.
Other than that, always escape strings:
Keywords LIKE '%" . mysql_real_escape_string($search_each) . "%'";
And while you're at it, why not replace the mysql extension with mysqli and prepared statements